Diameter-volume Inequalities and Isoperimetric Filling Problems in Metric Spaces

In this article we study metric spaces which admit polynomial diameter-volume inequalities for k-dimensional cycles. These generalize the notion of cone type inequalities introduced by M. Gromov in his seminal paper Filling Riemannian manifolds. In a first part we prove a polynomial isoperimetric inequality for k-cycles in such spaces, generalizing Gromov’s isoperimetric inequality of Euclidean type. In a second part we exhibit a connection between the asymptotic rank of a metric space and its isoperimetric inequalities. Namely, we prove that a complete metric space with cone type inequalities admits sub-Euclidean isoperimetric inequalities above its asymptotic rank. In particular, we obtain that isoperimetric inequalities can be used to detect the asymptotic rank of such spaces. The results of the second part for example apply to simply connected metric spaces of non-positive curvature in the sense of Alexandrov.